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''Es ist passiert'', “it just sort of happened”, people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.

<i>Es ist passiert</i>, &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.

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| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]

| align="right" colspan="3" | &mdash; Robert Musil, <i>The Man Without Qualities</i>, [Mus, 34]

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Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion.  In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.</math>  Then we discover that the simple features <math>\{ x, y, z \}</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}</math> of other features <math>\{ u, v \}</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].</math>  It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.

Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z</math> which present themselves to be simple enough in their own right and which form a satisfactory, if temporary foundation to provide a basis for discussion.

A particular transformation <math>F : [u, v] \to [x, y, z]</math> may be expressed by a system of equations, as shown below. Here, <math>F</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),</math> where each component map in <math>\{ f, g, h \}</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.</math>

 

 

 

A particular transformation <math>F : [u, v] \to [x, y, z]</math> may be expressed by a system of equations, as shown below.&nbsp; Here, <math>F</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),</math> where each component map in <math>\{ f, g, h \}</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.</math>

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Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.

Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}</math> on which is founded another universe of discourse.&nbsp; Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.

===Analytic Expansions : Operators and Functors===

===Analytic Expansions : Operators and Functors===

<ul><li><b>NB.</b>&nbsp; There is one aspect of the preceding construction which remains especially problematic.&nbsp; Why did we define the operators <math>\mathrm{W}</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character?&nbsp; Clearly, not all of the operator maps <math>\mathrm{W}F</math> have equally good reasons for placing their values in differential stocks.&nbsp; The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps.&nbsp; By default, only those values in the same functional component can be brought into algebraic modes of interaction.&nbsp; Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting that logical circumstance into algebraic forms of application has not yet been taken up.</li></ul>

<ul><li><b>NB.</b>&nbsp; There is one aspect of the preceding construction which remains especially problematic.&nbsp; Why did we define the operators <math>\mathrm{W}</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character?&nbsp; Clearly, not all of the operator maps <math>\mathrm{W}F</math> have equally good reasons for placing their values in differential stocks.&nbsp; The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps.&nbsp; By default, only those values in the same functional component can be brought into algebraic modes of interaction.&nbsp; Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting that logical circumstance into algebraic forms of application has not yet been taken up.</li></ul>

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